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113
De
finit
e
int
egr
al
su
mmatio
n
2
,
1
=
1
+
1
+
..
+
1
=
n
n
(n+
1)
i
=
H
2
+
...
+
n
=
i
=
1
2
n
(n+
1
)
(2n+
1)
I
i
=
142
4
...
+
n
=
--
i
=
1
-Ri
-
1)
=
1
+
3
+
5
+
...
+
2n
-
1)
=
7
n
=
4
-
,
5
I
pr
o
o
f
:
(2i
-1
=
(i)
-
E
=
2)i)
-
n
=
2
.
A
-n
=
nin
-
n
=
n
!
1
+
2
+
3
+
4
+
5
+
....
=
"
-
z
(R
amanujan
Sum)
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Ar
ea
Find
the
ar
e
a
A
of
the
re
g
i
o
n
under
the
line
y
=
x+
1
·
abo
v
e
the
x-
a
x
i
s
·
be
t
w
e
e
n
x
=
0
and
x
=
3
Ide
a
:
we
divide
the
re
g
i
o
n
int
o
small
re
c
t
a
n
g
l
e
pieces
·
up
p
er
bound
·
lower
bound
·
ta
k
e
limit
y
·
P(
3
,
4)
3
p1
3
,
4)
-+
+
+
-
↓
I
N
*
I
-
-
is
Xi-l
E
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We
ta
k
e
a
par
titio
n
of
10
.
33
by
0
=
Xo
<
X
,
<
...
<X
n
+
1
<
Xn
=
3
.
To
mak
e
the
comput
ation
easier
,
we
ta
k
e
Xx
=
P
=
0
...
n
On
each
[X
ix
,
Xi]
,
the
lower
bound
is
ta
k
e
n
by
f(
x
it
)
,
i
.
e
.
fix
it
)
=
f(
x
)
fo
r
all
xe
[
X
i
t
.
Xi
]
.
The
appr
o
ximation
fr
om
be
low
is
↳
=
E
,
f(
x
i
-
-
=
E
,
/X
i
+1
)
=
E(
+
1)
=
((
:
-)
+
=
i) +
.
n
=
-
-
4
+
3
=
2
+
3
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On
(xix
.
xi]
,
the
up
p
er
bound
is
ta
k
e
n
by
f(
x
i)
,
i
.
e
.
f(
x
)
<
fix
i)
fo
r
all
x=
(
X
i
t
,
xi
]
.
Un
=
fIXi)
-
=
Xit
=
E
,
(2
+
1)
=
i)
+
E
=
.
int
+
2
.
n
=
=
.
+
3
r
L
<
Ar
ea
<
Un
i
I
=
+
3
>
Ar
ea
-I
+
3
we
ta
k
e
the
limit
+
to
,
lim
L
Ar
ea
<
lim
Un
n
+
i
n
+
-
Ar
ea
=
9 +
3
.
I
1
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In
gener
al
,
fo
r
a
continuous
funct
ion
y
=
fix
)
on
[a
,
b]
,
·
we
ta
k
e
par
titio
n
P
a
=
Xo
<
X
,
<
...
<X
n
+
Xn
=
b
De
not
e
&x
i
=
Xi
-
Xi
-
1
·
On
ea
ch
piece
[xit
,
Xi]
ther
e
is
a
minimal
po
int
:
and
a
maximal
po
int
U
;
10
flil<
fix
)
<f
(Ui
)
0x
=/X
ix
.
Xi)
We
de
fine
217
,
P)
=
=
,
fli)
ox
:
lower
sum
01
,
>
=
E
,
finis
o
x
:
up
p
er
sum
RC
,
P
.
=
E
,
flci)
o
x
:
Riemann
su
m
Ci
=
SX
it
,
Xi]
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Then
we
alw
ay
hav
e
<
.p
)
=
R1t
,
P
.
c)
=
Ult
.
P)
.
Le
t
hips
be
the
number
of
par
titio
ns
de
f
.
and
1P
11
=
ma
x
/co
xi
1
i
<n
(
p
)
De
f
.
Le
t
f(
x
)
be
a
funct
ion
on
int
er
v
al
(a
,
b)
If
lim
Lif
.
p)
=
lim
UC
.
P)
fo
r
par
titio
n
n(p)
->
A
n(p)
-
0
11P11
->
0
11
P11
-
0
-
P
of
(a
.
b]
with
conditions
n(p)
->
+0
and
/P
1-
0
,
then
we
say
that
fix
e
is
int
egr
able
on
[a
,
33
,
and
we
de
not
e
the
limit
by
I
,
"f
ix
d
x
It
fo
l
l
o
w
s
that
Saf
ix
d
x
=
lim
RA
.
P
.
23
.
n(p)
+
Il
P11
-
0
-
Theor
em
If
f
is
continuous
on
[a
,
b]
then
f
is
int
egr
able
on
[a
.
b3
.
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RIGHTS
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Mor
e
on
the
symb
o
l
/f
ix
d
x
I
It
is
a
number
!
10
J
is
the
int
egr
al
sig
n
,
it
is
-m
any
su
m
-
I
2
We
ma
k
e
conv
ention
that
I
=-
1
!
30
The
funct
ion
o
is
called
the
int
egr
and
4
X
is
th
e
va
r
i
a
b
l
e
of
the
int
egr
and
5
Ex
is
the
diff
e
r
e
nt
ial
of
X
00
S
,
"
fix
dx
=
1
?
fit
s
&t
dum
m
y
va
r
i
a
b
l
e
7
we
als
o
wri
t
e
I
fix
s
do
x
as
Sid
xf
ixs
8
:
In
gener
al
,
we
wil
l
No
t
use
abo
v
e
de
finit
ion
fo
r
th
e
comput
ation
.
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Theor
em
3
(P
r
o
per
ties
of
de
finit
e
int
egr
al)
Le
t
f
and
9
be
int
egr
able
on
an
int
er
v
al
cont
aining
a
,
b
and
c
las
1
,
fix
idx
=
0
163
S
,
"
fix
s
dx
=
-1
,
fix
)
dx
Ja
=-
J
1
CLinearity)
Le
t
A
,
B
be
const
ants
,
th
en
((A f
ix
+
Bg
xs)d
x
=
Al
fix
dx
+
BS
!
gix
dx
id
s
!
fix
dx
+
)
;
fix
dx
=
Sc
fix
&x
(e)
If
a
b
and
fix
e
<
gix
s
fo
r
as
X2
b
,
them
(f
ixsax
?
gix
,
dx
1)
Tr
i
a
n
g
l
e
inequality)
((f
ix
d
x)</
-
/1x >1d
x
symmetric
pr
o
per
ties
19
1
Le
t
aso
and
fix
s
be
n
on
<-a
,
as
[
the
n
1 f
ix
d
x
=
0
·
fl
-
x)
=
-
f(
x
)
,
fl0)
=
0
.
14
>
Le
t
aso
and
fix
s
be
ti
on
on
S-a
,
as
the
n
6
+1
x
Ax
=
2)"
fix
dx
·
f(
-
x1
=
fi
x)
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Ex
ample
11
(
(x
+
1)d
x
=
So
xd
x
+
J
!
1d
x
=
+
3
MJ3
xd
x
=
(0x
d
x
+
13
x
d
x
y
=
So
x
d
x
+
1
.
!
+
d
+
=
0
en
writ
e
X
=
-
t
I
X
sin
ce
-
ru
n
s
fr
om
-3
to
0
then
X
ru
n
s
fr
o
m
3
to
0
(
-
%
+
dt
=
fix
x
)
d(
x)
=
+y
,
0x
d
x
=
-
!
xd
x
1)
S
!
*
&x
no
meaning
The
int
egr
ation
is
impr
oper
si
n
ce
the
funct
ion
*
has
no
de
finit
ion
fo
r
x
=
0
.
fix
s
=
*
is
an
odd
funct
ion
on
Df
=
/R1/03
de
le
t
e
origin
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Theor
em
(mean-
v
alue
Theor
em
fo
r
int
egr
als)
Le
t
f
be
continuous
on
a
closed
int
er
v
al
[a
.
b]
,
then
ther
e
ex
i
s
t
a
po
int
(-
[9
,
33
such
that
(f
ix
d
x
=
(b
-
a) f(x)
.
Pr
oof
Recall
the
Int
e
r
m
e
diat
e
-
Va
l
u
e
Th
e
o
r
e
m
:
If
gix
is
continuous
on
Sa
,
b]
,
then
it
ta
k
e
s
all
va
l
u
e
s
be
t
w
e
e
n
it
s
min
imal
va
l
u
e
an
d
it
s
ma
x
i
mum
va
l
u
e
·
Now
&
obt
ain
s
its
minimal
va
l
u
e
m
(at
x
=
1)
and
it
s
maximu
m
va
l
u
e
in
(at
x
=
U
,
i
.
e
.
M
=
fine
)
.
m(b
-
a)
<
(af
(x)
ax
-
>
M(
b
-
a)
i
.
e
.
fie
l
=
Saf
atix d
x
<
f(
u)
.
By
the
Int
e
r
m
e
di
at
e
-
va
l
u
e
Theor
em
,
fix
e
must
ta
k
e
on
eve
r
y
va
l
u
e
be
t
w
e
e
n
fle
)
and
fius
.
So
ther
e
is
so
me
po
in
t
be
t
w
e
e
n
1
and
a
such
that
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fx
x
=
(a
fix
)
dx
-
we
ca
l
l
it
the
-
me
a
n
va
l
u
e
of
we
re
w
r
i
t
e
it
as
f
on
Ca
.
b3
.
Saf
ix
d
x
=
(b
-
a)
f(
C)
.
I
De
finit
e
int
egr
als
of
piecewise
continuous
funct
ions
.
Le
t
Co<
C
,
<
...
<Cn
on
IR
A
funct
ion
f
de
fine
d
on
[Co
,
Cn
]
exc
e
p
t
pos
si
bly
at
so
me
of
those
Ci
,
is
called
pi
ec
ewi
se
cont
inuous
if
fo
r
eac
h
:
(1
:
<n
)
ther
e
ex
i
s
t
s
a
continuous
funct
ion
Fi
on
th
e
closed
set
Sit
,
(i
)
such
that
f(
x
)
=
F
:
on
op
e
n
set
(Cit
.
Ci)
.
Now
I"
fix
&x
t
I
.
Fix
edx
Y
=
I
is
NO
T
piecewis
continuous
on
St
.
12
.